AUSTRIAN-POLISH 1987 Problem 3

A function $f: R \to R$ satisfies $f (x + 1) = f (x) + 1$ for all $x$ . Given $a \in R$ , define the sequence $(x_n)$ recursively by $x_0 = a$ and $x_{n+1} = f (x_n)$ for $n \ge 0$ . Suppose that, for some positive integer m, the difference $x_m - x_0 = k$ is an integer. Prove that the limit $\lim_{n\to \infty}\frac{x_n}{n}$ exists and determine its value.